Machine Design Databook Episode 3 part 5 doc

Source: MACHINE DESIGN DATABOOK CHAPTER 24 MISCELLANEOUS MACHINE ELEMENTS2,3 24.1 CRANKSHAFTS2,3 SYMBOLS A b c d de dm E F Fc Fcomb Fic Fr F G h i0 ¼ lo ratio of length to diameter of crank Di Do ratio of inner to outer diameter of a hollow shaft I K¼ Kb Kt l le Mb area of cross section, m2 (in2 ) width of crank cheek, m (in) distance from the neutral axis of section to outer fiber, m (in) diameter (also suffixes), m (in) equivalent diameter, m (in) diameter of crankpin, m (in) diameter of main bearing, m (in) modulus of elasticity, GPa (psi) force acting on the piston due to steam or gas pressure corrected for inertia effects of the piston and other reciprocating parts, kN (lbf ) the component of force F acting along the axis of connecting rod, kN (lbf ) combined force, kN (lbf ) magnitude of inertia force due to the weight of connecting rod itself, kN (lbf ) total radial force acting on the crankpin, kN (lbf ) total tangential force acting on the crankpin, kN (lbf ) modulus of rigidity, GPa (psi) thickness of cheek or web (also with suffixes), m (in) moment of inertia, m4 , cm4 (in4 ) numerical combined shock and fatigue factor to be applied to the computed bending moment numerical combined shock and fatigue factor to be applied to the computed twisting moment length (also with suffixes), m (in) equivalent length, m (in) bending moment, N m (lbf in) 24.1 Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website MISCELLANEOUS MACHINE ELEMENTS 24.2 Mt p r Z   CHAPTER TWENTY-FOUR twisting moment, N m (lbf in) allowable pressure, MPa (psi) radius, throw of crankshaft, m (in) section modulus, m3 , cm3 (in3 ) normal stress (also with suffixes), MPa (psi) shear stress, MPa (psi) SUFFIXES b c comb e m max r rh t s  bending compressive combined elastic main maximum radial resultant in arm resultant in hub torque shaking tangential Other factors in performance or special aspects which are included from time to time in this chapter and are applicable only in their immediate context are not given at this stage Particular Formula FORCE ANALYSIS (Fig 24-1) The radial component of force Fc acting along the axis of connecting rod (Fig 24-1) The tangential component of force Fc acting along the axis of connecting rod (Fig 24-1) The radial component of force Fic (Fig 24-1) F Fc1 ẳ Fc cos ỵ ị ẳ s cos ỵ ị   sin  1À n0 ð24-1Þ F Fc2 ẳ Fc sin ỵ ị ẳ s sin ỵ ị   sin  n0 24-2ị Fic1 ẳ Fic cos 24-3ị where ẳ angle between the force Fic and the radial component of Fic The tangential component of force Fic (Fig 24-1) Fic2 ¼ Fic sin ð24-4Þ The total radial force acting on the crank Fr ẳ Fic1 ặ Fc1 24-5ị Fr ẳ Fic cos ặ Fc cos ỵ ị 24-6ị Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website MISCELLANEOUS MACHINE ELEMENTS MISCELLANEOUS MACHINE ELEMENTS Particular Formula F ¼ Fic2 Ỉ Fc2 The total tangential force acting on the crank ẳ Fic sin ặ Fc sin ỵ ị q Fcomb ẳ F2 ỵ F2 r  Fcomb c1 F θ α r lo Fs (b) Shaking force diagram Fc φ Fic1 Fc F 3l l d1 h c2 x c1 x -x l1 Arm F l ð24-8Þ b1 = 1.5 d0 b2 = 1.35 d l0 = 1.25 d0 Fr = Fic1 – Fc1 Fi r Fc2 Fic2 Fic γ ð24-7Þ Neutral axis of arm h1 r1 The resultant force on the crankpin (θ + φ) 24.3 b1 b2 x dm d2 = 1.75 d d1 = d0 h1 = 1.4 d0 h2 = 1.4 d d2 (a) FIGURE 24-1 (a) Forces acting on crankshaft (b) Vector sum of F and Fr lm h2 FIGURE 24-2 Overhung built-up crank SIDE CRANK Crankpin The maximum bending moment on the crankpin (Fig 24-2)  Mbmaxị ẳ Fcomb lo t ỵ 2  24-9ị ẳ Fcomb l lo ỵ c2 ẳ distance from centroidal axis to the application of load (Fig 24-2), m (in) s 32lFcomb 24-10ị ẳ b where l ¼ The crankpin diameter with respect to the bending moment where b ¼ allowable bending stress, MPa (psi) The diameter of crankpin from the consideration of bearing pressure From Eqs (24-10) and (24-11) neglecting t=2 and eliminating lo the equation for crankpin diameter Empirical relation to determine the length of crankpin Fcomb lo p sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 16Fcomb ẳ pb 24-11ị lo ẳ i0 d o 24-13ị ¼ where i0 ¼ lo ¼ 1:25 to 1.5 Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website ð24-12Þ MISCELLANEOUS MACHINE ELEMENTS 24.4 CHAPTER TWENTY-FOUR Particular Another relation for the crankpin length/diameter ratio Another relation for the crankpin diameter Formula l i ¼ o ¼ sffiffiffiffiffiffiffiffiffiffiffi 0:2b p sffiffiffiffiffiffiffiffiffiffiffi Fcomb ¼ i0 p ð24-14Þ ð24-15Þ HOLLOW CRANKPIN The crankpin length/diameter ratio l i ¼ o ¼ Do sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0:2ð1 À K4 Þ p where K ẳ The crankpin outside diameter 24-16ị Di Do s Fcomb Do ẳ i0 p 24-17ị Crank arm CRANK ON HEAD-END DEAD-CENTER POSITION When the crank is on the head-end dead-center position, the section XX (Fig 24-2) of the arm is subjected to bending moment Mb ¼ Fcomb  l ð24-18Þ The direct compressive stress due to the load Fcomb (i.e., more specifically by its component Fc ) c ẳ Fcomb A 24-19ị The resultant stress in the crank arm at XX ra ẳ Fcomb MbC ặ A I where 24-20ị A ẳ area of cross section of the arm at XX, m2 (in2 ) c ¼ distance from the neutral axis of section to outer fiber of arm, m (in) I ¼ moment of inertia of the section, cm4 (in4 ) CRANK ON CRANK-END DEAD-CENTER POSITION The direct tensile stress in the plane of the hub of crankshaft section passing through the shaft center due to load Fcomb (Fig 24-2) t ¼ The bending stress in the section due to bending moment Fcomb  a b ¼ Fcomb h2 ðd2 À dÞ Fcomb  a Z where Z ¼ section modulus, cm3 (in3 ) Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website ð24-21Þ ð24-22Þ MISCELLANEOUS MACHINE ELEMENTS MISCELLANEOUS MACHINE ELEMENTS Particular 24.5 Formula r ẳ t ặ b 24-23ị The bending moment in the plane of rotation of the crank Mb ¼ Fcomb  l 24-24ị The bending stress b ẳ Mb c1 Zb 24-25ị The torsional moment Mt ẳ Fcomb r1 24-26ị The shear stress ¼ Mt c Zt qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i h max ẳ b ỵ 2 ỵ 4 b ð24-27Þ The resultant stress in the plane of the hub of crankshaft section passing through the shaft center CRANK PERPENDICULAR TO THE CONNECTING ROD The maximum normal stress for crank made of cast iron 24-28ị q 2 ỵ 4 b ð24-29Þ The shaking force on the main bearing from F and Fr (Fig 24-1b) Fs ¼ vector sum of F and Fr ð24-30Þ The diameter of main bearing taking into consideration the bearing pressure on the projected area of the crankshaft dm ¼ The maximum shear stress for the crank made of steel max ¼ DIMENSION OF CRANKSHAFT MAIN BEARING (Fig 24-2b) Fs lm p 24-31ị where lm ẳ length of bearing, m (in) p ¼ allowable bearing pressure, MPa (psi) The bending movement on the crankshaft Mb ẳ Fcomb l1 24-32ị lo l þ h2 þ m 2 where l1 ¼ h2 ¼ hub length, m (in) lo ¼ length of crankpin, m (in) lm ¼ length of bearing on crankshaft, m (in) The torque on the crankshaft Mt ¼ Fcomb  r The diameter of crankshaft taking into consideration indirectly the fatigue and shock factors where r ¼ throw of the crank, m (in) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 16 dm ¼ Kb Mb ỵ Kb Mb ị2 ỵ Kt Mt ị2 e ð24-34Þ Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website ð24-33Þ MISCELLANEOUS MACHINE ELEMENTS 24.6 CHAPTER TWENTY-FOUR Particular Formula The length of main bearing lm ¼ Fs dm p ð24-35Þ PROPORTIONS OF CRANKSHAFTS For proportions of crankshaft Refer to Figs 24-2 to 24-10 0.8 to 1.1do 0.5 to 0.9do d0 1.4d0 Throw 1.1d1 d 1.8d1 d1 K to 2.5d FIGURE 24-3 Overhung built-up crank 1.1 to 1.2d FIGURE 24-4 Overhung forged crank Fcomb L lo Center of bearing dm h lm 12mm c 12mm lm b FIGURE 24-6 Center crank (American Bureau of Shipping method) dc FIGURE 24-5 Disk crank h Dc h r= d1 d2 d h1 Center of bearing dm h2 t D do L d3 dj r I Dj dj Ie e FIGURE 24-7 Equivalent length of crankshaft h c h e FIGURE 24-8 Center hollow crank Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website b MISCELLANEOUS MACHINE ELEMENTS MISCELLANEOUS MACHINE ELEMENTS Particular Formula B = Cylinder bore 0.71 to 0.80B 0.5d R = 0.35B 0.7B + 9.5 = d 0.06 to 0.12B 24.7 0.58B 0.53 31B + 9.5 to 0.6B 0.825B + All dimensions in mm FIGURE 24-9 Empirical proportion for center crank 2.35d 1.5d 3.25d 0.580d 3.75d d d 2.25d 3.75d 2.15d 1.15d 0.58d Throw d 1.5d 2.1d CENTER CRANK (Fig 24-6) Crankpin The maximum bending moment treating the crankpin as a simple beam with concentrated load at the center Fcomb lo ỵ h ỵ lm ị where lo ¼ length of crankpin, m (in) lm ¼ length of main bearing, m (in) h ¼ thickness of cheek, m (in) Mbc ¼ Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website ð24-36Þ MISCELLANEOUS MACHINE ELEMENTS 24.8 CHAPTER TWENTY-FOUR Particular The diameter of the crankpin based on maximum bending moment Mbc Formula sffiffiffiffiffiffiffiffiffiffiffiffiffi 32Mbc ¼ b 24-37ị where b ẳ design stress, MPa (psi) The diameter of crankpin based on bearing pressure between pin and the bearing ẳ Fcomb lo p 24-38ị Dimensions of main bearing Fcomb le 24-39ị where le ẳ equivalent length of crankshaft, m (in) The maximum bending moment treating the center crank as a simple beam with load concentrated at the center Mbb ¼ The twisting moment Mt ¼ Fcomb  r The diameter of crankshaft at main bearing taking into consideration the fatigue and shock factors sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  q 16 Kb Mbb ỵ Kb Mbb ị2 ỵ Kt Mt ị2 dm ẳ e 24-40ị 24-41ị The diameter of the crankshaft based on bearing pressure dm ¼ Fs lm p ð24-42Þ American Bureau of Shipping formulas for center crank The thickness h of the cheeks or webs (Fig 24-6) The diameter of crankpins and journals (Fig 24-6) h ¼ 0:4d to 0:6d sffiffiffiffiffiffiffiffiffi Dpc d¼a b 24-43ị 24-44ị where a ẳ coecient from Table 24-1A D ¼ diameter of cylinder bore, m (in) p ¼ maximum gas pressure, MPa (psi) c ¼ distance over the crank web plus 25 mm (1.0 in) (Fig 24-6) b ¼ allowable fiber stress, MPa (psi) bh2 ! 0:4d3 ð24-45aÞ b2 h ! d3 The thickness h and the width b of crank cheeks must satisfy the conditions ð24-45bÞ Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website MISCELLANEOUS MACHINE ELEMENTS MISCELLANEOUS MACHINE ELEMENTS Particular 24.9 Formula EQUIVALENT SHAFTS A portion of a shaft length l and diameter d can be replaced by a portion of length le and diameter de The length he equivalent to crank web  le ẳ l de d  24-46ị rC B where 24-47ị he ẳ C ẳ 32 d4 G ¼ torsional rigidity of the crankpin e B ¼ 12 hb3 E ¼ flexural rigidity of the web The equivalent length crankshaft le of Fig 24-7 varies between 0:95l < le < 1:10l The equivalent length of commercial crankshaft for solid journal and crankpin according to Carter (Fig 24-8) Le ¼  d4 e  The equivalent length of commercial crankshaft for hollow journal and crankpin according to Carter (Fig 24-8) Le ¼ d4 e The equivalent length of crankshaft for solid journal and crankpin according to Wilson (Fig 24-8) Le ¼ The equivalent length of crankshaft for hollow journal and crankpin according to Wilson (Fig 24-8)  d4 e  Le ẳ d4 e e ỵ 0:8a 0:75b 1:5r ỵ ỵ D4 D4 ac c J 24-48ị  e ỵ 0:8a 0:75b 1:5r ỵ ỵ D4 À d4 D4 À d4 ac3 c c J J 24-49ị  24-50ị e ỵ 0:4DJ b ỵ 0:4Dc r 0:2DJ ỵ Dc ị ỵ ỵ D4 D4 ac3 c J  24-51ị  e ỵ 0:4DJ b ỵ 0:4Dc r 0:2DJ ỵ Dc ị ỵ ỵ D4 À d4 Dc À d4 ac3 c J J ð24-52Þ EMPIRICAL PROPORTIONS For empirical proportions of side crank, built-up crank, and hollow crankshafts Refer to Figs 24-2 to 24-10 The film thickness in bearing should not be less than the values given here for satisfactory operating condition: Main bearings h ¼ 0:0025 mm (0.0001 in) to 0.0042 mm (0.0017 in) 24-52aị Big-end bearings h ẳ 0:002 mm (0.00008 in) to 0.004 mm (0.00015 in) ð24-52bÞ The oil flow rate through conventional central circumferential grooved bearings Q¼ kpc3 d ð1 þ 1:5"2 Þ  L Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website ð24-52cÞ MISCELLANEOUS MACHINE ELEMENTS 24.10 CHAPTER TWENTY-FOUR Particular Formula where Q ¼ oil flow rate, m3 /s (gal/min) k ¼ a constant ¼ 0:0327 SI units ¼ 4:86  104 US Customary System Units p ¼ oil feed pressure, Pa (psi) c ¼ D À d ¼ diametral clearance, m (in)  ¼ absolute viscosity (dynamic viscosity), Pa s (cP) d ¼ bearing bore, m (in) L ¼ land width, m (in) " ¼ attitude or eccentricity ratio For oil flow rate in medium and large diesel engines at 0.35 MPa 0.5 psi Refer to Table 24-1B The velocity of oil in ducts on the delivery side of the pump v ¼ 1:8 to 3.0 m/s (6 to 10 ft/s) ð24-52dÞ The velocity of oil in ducts on the suction side of the pump v ẳ 1:2 m=s4 ft=sị 24-52eị The delivery pressure in modern high-duty engines p ¼ 0:28 to 0.42 MPa (40 60 psi) 24-52f ị pmax ẳ 0:56 MPa 80 psiị ð24-52gÞ For housing tolerances Refer to Table 24-1C TABLE 24-1A Coefficient a in the American Bureau of Shipping formula [Eq (24-44)] Ratio of stroke to distance over crank webs ¼ l=c Number of cylinder Type Four-stroke Two-stroke 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 Explosion engines 1, 2, 3, 5, 10, 11, 12 1, 2, 3, 5, 12 16 1, 5, 1, 1.17 1.17 1.17 1.18 1.17 1.19 1.20 1.22 1.25 1.17 1.17 1.19 1.20 1.19 1.22 1.24 1.25 1.29 1.17 1.17 1.21 1.23 1.22 1.25 1.27 1.29 1.33 1.17 1.17 1.23 1.25 1.25 1.28 1.30 1.32 1.36 1.17 1.19 1.25 1.28 1.28 1.32 1.33 1.36 1.40 1.17 1.20 1.28 1.31 1.31 1.35 1.37 1.39 1.44 1.17 1.22 1.30 1.33 1.34 1.38 1.40 1.42 1.47 1.17 1.24 1.32 1.35 1.36 1.41 1.43 1.45 1.50 Air-injection diesel engines 5, Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website MISCELLANEOUS MACHINE ELEMENTS 24.26 CHAPTER TWENTY-FOUR 24.4 PISTON AND PISTON RINGS SYMBOLS2,3 A b B c C d dg D Dr E F F h h1 , h2 , h3 H l g i0 ¼ dg lg L Mb n Pb p r R, Ri , Rh th tr Tc Te w area of cross section of piston head, m2 (in2 ) width of face of piston, m (in) diameter of bore, m (in) heat-conduction factor, kJ/m2 /m/h/K (Btu/in2 /in/h/8F) higher heat value of fuel used, kJ/kg (Btu/lb) nominal diameter of piston ring, m (in) diameter of piston rod, m (in) diameter of gudgeon pin, m (in) diameter of bore (cylinder), m (in) root diameter of the piston ring groove, m (in) modulus of elasticity, GPa (psi) force, kN (lbf ) diametral load on the piston ring to close the gap which is less than 2.45 N (0.55 lbf ) tangential load on the piston ring to close the gap which is less than 2.45 N (0.55 lbf ) thickness (also with subscripts), m (in) radial thickness of piston ring, m (in) thickness as shown in Fig 24-24, m (in) heat flowing through the head, kJ/h (Btu/h) length/diameter ratio length of gudgeon pin, m (in) length of piston, m (in) bending moment, N m (lbf in) safety factor brake horsepower (bhp) pressure, MPa (psi) radius, m (in) radius as shown in Fig 24-22b, m (in) thickness of head, m (in) thickness under ring groove, m (in) temperature at center of head, 8C (8F) temperature at edge of head, 8C (8F) weight of fuel used, kg/bhp/h axial width of ring, m (in) stress (also with subscripts), MPa (psi) angle (Fig 24-23), deg   TABLE 24-4 Dimensions of cast-iron piston up to 430 mm diameter (Fig 22-24) (all dimensions in cm) Diameter of cylinder, D Diameter of piston rod, d d1 b a h h1 h2 h3 d2 d3 15.0 20.0 25.0 30.0 35.0 40.0 2.8 3.4 4.0 5.0 5.6 5.9 2.5 3.1 3.7 4.7 5.0 5.6 7.5 8.1 9.0 10.0 11.2 11.8 1.2 1.4 1.6 1.7 1.9 1.9 1.2 1.2 1.4 1.6 1.6 1.7 1.4 1.6 1.7 1.7 1.9 2.2 0.8 0.95 0.95 1.2 1.2 1.2 0.8 0.8 0.95 0.95 0.95 0.95 3.1 3.4 4.0 4.6 5.3 5.8 2.5 3.1 3.7 4.7 5.0 5.6 Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website MISCELLANEOUS MACHINE ELEMENTS MISCELLANEOUS MACHINE ELEMENTS Particular 24.27 Formula STEAM ENGINE PISTONS Piston rods The diameter of piston rod dẳD r p a 24-104ị where The diameter of piston rod according to Molesworth p ¼ unbalanced pressure or difference between the steam inlet pressure and the exhaust, MPa (psi) u ¼ allowable stress, MPa (psi) a ¼ n Note: a is based on a safety factor of 10 for doubleacting engines and for single-acting engines (The diameter of piston rod is usually taken as to the diameter of the piston.) pffiffiffi d ¼ 0:0044D p for cast-iron pistons 24-105ị p d ẳ 0:00338D p for steel pistons ð24-106Þ Ri R Rh (a) θ (b) FIGURE 24-23 Conical plate piston FIGURE 24-22 Plate piston h2 h1 d2 a b a h3 d1 h d3 d D FIGURE 24-24 Cast-iron piston of diameter 400 mm (16.0 in) Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website MISCELLANEOUS MACHINE ELEMENTS 24.28 CHAPTER TWENTY-FOUR Particular Formula PROPORTIONS FOR PRELIMINARY LAYOUT FOR PLATE PISTONS Box type (Figs 24-22a and 24-24) Width of face b ¼ 0:3 to 0:5D ð24-107Þ Thickness of walls and ribs for low pressure p h ẳ 2R ỵ 50 cmị 0:003 p ỵ 0:0275 cmị 24-108ị or hẳ The thickness of walls and ribs for high pressure For dimensions of conical plate piston For dimensions of cast-iron piston of diameter 2R ỵ 10 mm (0.40 in) 60 24-109ị hẳ 2R ỵ 10 mm (0.40 in) 40 ð24-110Þ Refer to Fig 24-23 400 mm Refer to Fig 24-24 and Table 24-4 Disk type (Fig 24-22b) Width of face b ¼ 0:3 to 0:5D Thickness of walls and ribs for low pressure pffiffiffi h ẳ 2R ỵ 12:5 cmị 0:0096 p ỵ 0:057 cmị 24-112ị 24-111ị The hub thickness h1 ẳ 0:45d 24-113ị The hub diameter Dh ¼ 2Rh ¼ 1:6  the piston diameter 24-114ị Width of piston rings w ẳ 0:03D to 0:06D 24-115ị Thickness of piston rings h ẳ 0:025D to 0:03D ð24-116Þ For dimensions of cast-iron piston Refer to Table 24-4 STRESSES (a) Distributed load over the plate inside the outer cylindrical wall (i.e., the area R2 ) i (1) Stress at the outer edge (Fig 24-22b) (2) Stress at the inner edge (Fig 24-22b) 1 ¼   3p 4R2 R R2 3R2 ỵ h ln i i h Ri À Rh Rh 4h2 ð24-117Þ 2 ẳ   3p 4R2 R2 R R2 ỵ R2 À i h2 ln2 i i h Rh Ri À Rh 4h2 ð24-118Þ Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website MISCELLANEOUS MACHINE ELEMENTS MISCELLANEOUS MACHINE ELEMENTS Particular (b) Load on the outer wall, pðR2 À R2 Þ distributed i around the edge of the plate (1) Stress at the outer edge (Fig 24-22b) (2) Stress at the inner edge (Fig 24-22b) 24.29 Formula   3pðR2 À R2 Þ 2R2 Ri i h 3 ¼ ln 1À Ri À R2 Rh 2h2 h   3pðR2 À R2 Þ 2R2 Ri i i ln 1À 4 ¼ Ri À R2 Rh 2h2 h ð24-119Þ ð24-120Þ (3) The sum of the stresses at the outer edge o ẳ 1 ỵ 3 24-121ị (4) The sum of the stresses at the inner edge i ẳ 2 ỵ 4 24-122ị (Note: o or i should not be greater than the permissible stress of the material A safety factor, n, of can be used.) Dished or conical type (Fig 24-23) An empirical formula for the thickness of conical piston (Fig 24-23) pffiffiffiffiffiffiffiffiffiffiffiffi h ¼ 0:288 pD= sin  SI ð24-123aÞ where p and  in MPa, and D and h in m pffiffiffiffiffiffiffiffiffiffiffiffi h ¼ 9:12 pD= sin  Customary Metric ð24-123bÞ where p and  in kgf/mm2 , D and h in mm p h ẳ 1:825 pD= sin  USCS 24-123cị where p and  in psi, D and h in in The height of boss H ¼ 1:1K The diameter of boss Dh ẳ 1:7K for small pistons 24-124ị 24-125aị Dh ¼ 1:5K for large pistons and light engines ð24-125bÞ The thickness h1 measured on the center line h1 ¼ Kc 24-126ị where c ẳ to 0.75 depending on the angle of inclination  (Refer to Table 24.5.)  ¼ varies from 68 to 358 K ¼ to 4.5 for varying pressure and diameter Also refer to Table 24-6 for values of K For calculating hub diameter, width of piston rings, and thickness of piston rings Refer to Eqs (24-114) to (24-116) Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website MISCELLANEOUS MACHINE ELEMENTS 24.30 CHAPTER TWENTY-FOUR Particular Formula PISTONS FOR INTERNAL-COMBUSTION ENGINES Trunk piston (Fig 24-25) The head thickness of trunk pistons (Fig 24-25a) sffiffiffiffiffiffiffiffiffiffiffiffi 3PD2 th ẳ 16 24-127ị where  ẳ 39 MPa (5.8 kpsi) for close-grained cast iron ¼ 56.4 MPa (8.2 kpsi) for semisteel or aluminum alloy ¼ 83.4 MPa (12.0 kpsi) for forged steel COMMONLY USED EMPIRICAL FORMULAS IN THE DESIGN OF TRUNK PISTONS FOR AUTOMOTIVE-TYPE ENGINES th ¼ 0:032D þ 1:5 mm Thickness of head (Fig 24-25a) th ¼ 0:00D ỵ 0:06 in th ẳ D tr USCS 24-128aị SI ð24-129aÞ where tr th H 0:194cðTc À Te Þ 24-128ị HD2 0:16cTc Te ịA ẳ The head thickness for heat flow SI Tc À Te ¼ 2058C (4008F) and Tc ¼ 698K, 4258C (8008F) for cast-iron piston ÁT ¼ Tc À Te ¼ 558C (1308F) and Tc ¼ 533K, 2608C (5008F) for aluminum piston (a) c ¼ 2.2 for cast iron Dr c ¼ 7.7 for aluminum tr th ¼ (c) (b) HD2 H ¼ 16cÁTA 12:5cÁT USCS ð24-129bÞ (d) D FIGURE 24-25 Trunk piston for small internal-combustion engine (a) piston laid out for heat transfer; (b) piston modified for structural efficiency; (c and d) alternate pin designs Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website MISCELLANEOUS MACHINE ELEMENTS 24.31 MISCELLANEOUS MACHINE ELEMENTS Particular Formula ð24-130Þ The thickness of wall under the ring groove tr ¼ thickness of head ¼ th pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tr ¼ Dr ặ D2 4Dth ị r The heat ow through the head H ẳ KCwPb 24-132ị Thickness of wall under the ring (Fig 24-25a and b) ð24-131Þ where w ¼ weight of fuel used, kJ/kW/h (lbf/bhp/h) K ¼ constant representing that part of heat supplied to the engine which is absorbed by the piston ¼ 0.05 (approx.) Pb ¼ brake horsepower per cylinder The root diameter of ring grooves, allowing for ring clearance Dr ¼ D À 2w ỵ 0:006D ỵ 0:02 inị USCS Dr ẳ D 2w ỵ 0:006D ỵ 0:5 mmị at the compression rings Dr ẳ D 2w ỵ 0:006D ỵ 1:5 mmị SI 24-132aị SI 24-132bị Dr ẳ D 2w þ 0:00D þ 0:06 inÞ at the oil grooves USCS where Dr and D in mm (in) Length L of piston L ¼ D to 1:5D For chemical composition and properties of aluminum alloy piston ð24-133Þ Refer to Table 24-10B Gudgeon pin The diameter of gudgeon pin sffiffiffiffiffi F dr ¼ i0 p ð24-134Þ where F ¼ maximum gas pressure corrected for inertia effect of the piston and other reciprocating parts, kN (lbf ) p ¼ working bearing pressure ¼ 9.81 MPa (1.42 kpsi) to 14.7 MPa (2.13 kpsi) lg ¼ 1:5 to dg The length/diameter ratio of gudgeon pin i0 ¼ For gudgeon pin allowable oval deformation Refer to Fig 24-28c For empirical relations and proportions of pistons Refer to Figs 24-26 to 24-28a Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website ð24-135Þ MISCELLANEOUS MACHINE ELEMENTS CHAPTER TWENTY-FOUR 0.2B 24.32 0.11B X 0.052B 7D 0.35B=D Local clearance 0.05D X 1.25B 0.67B 0.1B 0.4B 0.22B 0.04B 0.0 42B 0.07B 0.37B Local clearance 0.1B 0.035B B = cylinder bore FIGURE 24-26 Proportions of a typical alloy piston A 0.2B 0.03 to 0.05B to 0.2 B 0.3 Taper Offset 0.375 to 0.4B 1.75 to 2.0B C 0.025 to 0.033B 0.170 to 0.125B B 1.0B 0.2 to 0.25B B (Nominal) 0.03 to 0.04B D FIGURE 24-27 Proportions of an iron piston Fig 24-27 mm (in) Cylinder 152.5 bore (6) Crown A 16 thickness (5/8) Clearance B 0.760 (0.03) Clearance C 0.125 (0.005) Clearance D 0.125 (0.005) mm (in) mm (in) mm (in) mm (in) 203.2 (8) 19 (3/4) 0.900 (0.035) 0.225 (0.008) 0.150 (0.006) 254 (10) 32 (11) 1.145 (0.045) 0.230 (0.009) 0.180 (0.007) 305 (12) 41.5 (15) 1.525 (0.06) 0.255 (0.01) 0.200 (0.008) 406.5 (16) 47.5 (17) 2.030 (0.08) 0.255 (0.01) 0.230 (0.009) Piston weight ¼ 40:715B3 N (approx.) SI where B ¼ cylinder bore, m Piston weight ¼ 0:15B3 lbf USCS where B in in Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website MISCELLANEOUS MACHINE ELEMENTS MISCELLANEOUS MACHINE ELEMENTS Ra d 2.2B B.less 1000 d 0.11B 0.085B B B.less 1000 B.less 3B 0.045B 4000 0.4B 0.045B 0.425B 1.35 to 1.5B 0.8 to 0.95B 0.22B 0.03B Ra Relief in way of pin bosses B = 1000 24.33 0.036B 0.21B 0.03B 3B B.less 4000 650 d 90 600 550 80 A 450 48 400 σ = 0.415P /m G /m MN 300 250 200 0.100 0.150 0.200 d t ,P= GAS LOAD AREA A MN 34 350 70 55 2) n f/i IN lb n ) AR 00 f/i 2) BE lb 00 n (1 000 lbf/i ) SS n BO (9 f/i ) N/ 00 lb PI (8 n N f/i 00 EO /m lb 2) (7 DG n MN GU f/i 00 /m 69 lb (6 MN /m 62 00 MN (5 /m MN 500 0.250 0.300 t GUDGEON PIN d PR 60 50 ES SU RE ’P’ 0.350 40 0.400 GUDGEON PIN FATIGUE STRESS,σ thousands of × 103 100 t 41 GUDGEON PIN FATIGUE STRESS,σ MN/m2 700 lbf/in2 FIGURE 24-28(a) Iron piston for small engines (B ¼ cylinder bore) 30 0.450 FIGURE 24-28(b) Fatigue stress in gudgeon pins for various pin and piston geometries (M J Neale, Tribology Handbook, Butterworth-Heinemann, 1973.) Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website MISCELLANEOUS MACHINE ELEMENTS 24.34 CHAPTER TWENTY-FOUR Formula CYLINDER BORE DIAMETER (NOMINAL), in 10 11 12 13 14 15 16 17 18 60 PIN OVAL DEFORMATION δ = (a b) D2 P d m 171 L δ D2 P d ( OR t = 900 in) L δ t= DEFORMATION, µm 50 L t 40 d b a D DIA 30 P MAX PRESSURE MN/m2 (lbf/in2) 20 10 2.6 2.5 2.4 2.3 2.2 2.1 2.0 1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1.0 0.9 0.8 0.7 0.6 0.5 DEFORMATION, THOUSANDTHS OF AN INCH Particular 100 150 200 250 300 350 400 CYLINDER BORE DIAMETER (NOMINAL), mm FIGURE 24-28(c) Gudgeon pin allowable oval deformation (M J Neale, Tribology Handbook, Butterworth-Heinemann, 1973.) For fatigue stress in gudgeon pins Refer to Fig 24-28b For empirical proportions and values of cylinder cover, cylinder liner, and valves Refer to Figs 24-30 to 24-33 Piston rings D for concentric rings 20 D opposite the joint of eccentric rings 27:5 24-136bị wẳ For land width or axial width of piston ring (w) required for various groove depths (g) and maximum cylinder pressure, pmax w¼ w¼ Width of rings D at the joint of eccentric rings 55 Refer to Fig 24-28d Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website ð24-136aÞ ð24-136cÞ MISCELLANEOUS MACHINE ELEMENTS MISCELLANEOUS MACHINE ELEMENTS 24.35 Groove depth g, in 0.200 0.250 0.300 0.350 0.400 0.450 0.500 2400 2200 ALUMINIUM PISTON LAND WIDTH W TO DETERMINE LAND WIDTHS FOR PISTONS IN CAST IRON OR STEEL USE THE FOLLOWING CONSTANTS 16.0 15.0 0.600 1600 16 1400 0.550 1200 0.500 M 12 13 15 M N/ M m2 N/ m2 11 M N/m N/ M m2 65 N/ M m2 N /m STEEL W FROM GRAPH × 0.5 13.0 0.650 1800 CAST IRON W FROM GRAPH × 0.700 14.0 2000 12.0 11.0 1000 /m N M 0.450 0.350 W, in 9.0 W, mm 10.0 m MAX CYLINDER M PRESSURE, Ibf/in2 0.400 N/ 8.0 0.300 7.0 g w 6.0 0.250 5.0 0.200 4.0 0.150 3.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 Groove depth, mm FIGURE 24-28(d) The land width required for various groove depths and maximum cylinder pressures (M J Neale, Tribology Handbook, Butterworth-Heinemann, 1973.) Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website MISCELLANEOUS MACHINE ELEMENTS δf δc Fθ Fθ Free ring Compressed ring r v r + v+ d d φ δc = circumferential clearance gap δf = free piston ring gap h = radial depth or wall thickness of piston ring w = axial width of piston ring d = nominal diameter of piston ring r = radius of neutral axis of ring h w FIGURE 24-28(e) Nomenclature of piston ring and tangential force, F d Variable pv = p (φ) Constant pc Fd Fd φ pc FIGURE 24-28( f ) Typical variable and constant contact pressure distribution around piston rings for four-stroke engines (Courtesy: Piston Ring Manual, Goetze AG, D5093 Burscheid, Germany, August 1986.) FIGURE 24-28(g) Diametrically opposite force (Fd ) applied on piston ring Gap clearance Fθ Fθ FIGURE 24-29 Tangentially applied force, F , on a piston ring 24.36 Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website MISCELLANEOUS MACHINE ELEMENTS MISCELLANEOUS MACHINE ELEMENTS Particular 24.37 Formula   d À1 F 5:37 h E¼ w when the ring is diametrically loaded ð24-137aÞ   d 14:14 À F h E¼ w when the ring is tangentially loaded The modulus of elasticity of piston ring as per Indian Standards 24-137bị where E ẳ modulus of elasticity, MPa (psi)  ¼ difference between free gap and gap after applying the load, mm (in) The bending moment produced at any cross section of the ring by the pressure uniformly distributed over the outer surface of the ring at an angle  measured from the center line of the gap of the ring (Figs 24-28e and 24-28f ) Mb ¼ À2pwr2 sin2  Mb ẳ pwr2 ỵ cos ị 24-138aị 24-138bị where r ẳ radius of neutral axis, mm (in) p ¼ pressure at the neutral axis of the piston ring, Pa (psi) The bending moment (Mb ) in Eq (24-138a) in terms of tangential force, F The uniform contact pressure of the piston ring on the wall Mb ¼ F r1 ỵ cos ị pẳ En f 7:07dd=h 1ị3 24-138cị 24-138dị where d ẳ external piston ring diameter h ¼ radial depth or wall thickness of piston ring En ¼ nominal modulus of elasticity of material of the ring f ¼ free ring gap The radial distance from a point in piston ring to obtain a uniform pressure distribution (Fig 24-28e) according to R Munroa a ro ¼ r ỵ v ỵ dv In M J Neale, ed., Tribology Handbook, Section A31, Newnes-Butterworth, London, 1973 Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website ð24-138eÞ MISCELLANEOUS MACHINE ELEMENTS 24.38 CHAPTER TWENTY-FOUR Particular Formula where 067B+4 04B 05B 067B + 35.5 09 to 12B 065B All dimensions in mm FIGURE 24-30 Proportion for four-stroke cover, 100- to 450-mm bore (B ¼ cylinder bore) Fr4 ð1 À cos  ỵ  sin ị En I "   r Fr3 ð À  cos  sin ị dv ẳ 2 En I # vẳ 24-138f ị sin  ỵ  cos ị where F ẳ (mean wall pressure  ring axial width) x 0.1 to 0.08 to 0.15B 0.12B 2.5 0.75 0.08 to 0.12B 9.15 18D to 21D to 4.75 1.25 to 1.35B 1.18 to 1.2B 3.45 0.1 Clear on dia 0.75 to 1.0B 3.25 r ¼ radius of neutral axis, when the ring is in place inside the cylinder (Fig 24-28e) 25 to 30D 25D 0.03B 0.07to 0.08B 18 to 22lbf/sq in of valve area H J B 0.0012 38D 65D 7R 5B 8000 Clear on dia 7.5 G w 0.07B 0.05B F B 0.0125 X Lowest ring on BDC All dimensions in mm FIGURE 24-31 Empirical rules for average practice in liner design (B ¼ cylinder bore) D FIGURE 24-32 Valve seated directly in the cylinder head Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website MISCELLANEOUS MACHINE ELEMENTS MISCELLANEOUS MACHINE ELEMENTS Particular 24.39 Formula  ¼ angle measured from bottom of the vertical line passing through the center of the gap of the ring as shown in Fig 24-28e I ¼ moment of inertia of the ring The relation between the ratio of fitting stress ft to nominal modulus of elasticity (En ) in terms of h, d, and f ft 48h f ỵ 0:00dị ẳ En 3hd=h À 1Þ2 The relation between the ratio of working stress (w ) to nominal modulus of elasticity (En ) in terms of h, d, and f 4ðf À 0:00dÞ w ¼ En 3hðd=h À 1Þ2 The relation between the ratio of the sum of (ft ỵ w ) to nominal modulus of elasticity (En ) in terms of d and h ft ỵ w 32 ẳ En 3d=h 1ị2 where ft ¼ opening stress when fitting the piston ring onto the piston For preferred number of piston rings Refer to Table 24-10C For properties of typical piston ring materials Refer to Table 24-10D 0.27D off Rib 0.27D D D 2.0 0.16D D C E Two ribs thus (revolved section) Coreplug 0.12D Cooling water connection N B Two bosses thus 1.1D 24-138hị where w ẳ working stress when the piston ring is in the cylinder Equation (24-138i) is independent of f Hardened ð24-138gÞ Copper joint FIGURE 24-33 Valve with removable cage Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website ð24-138iÞ MISCELLANEOUS MACHINE ELEMENTS 24.40 CHAPTER TWENTY-FOUR Particular The circumferential clearance (c ) or gap between ends of ring Formula c ẳ dp T 24-138jị where p ¼ coefficient of expansion of piston ring material T ¼ operating temperature d ¼ cylinder diameter An expression for pressure acting on ring from Eqs (24-138b) and (24-138c) pẳ F rw 24-139aị The pressure in the radial outward direction against the cylinder pẳ 2F dw 24-139bị For variable and constant radial contact pressure distribution of piston ring Refer to Fig 24-28f The diametral load which acts at 908 to the gap required to close the ring to its nominal diameter, d (Fig 24-28g) Fd ¼ 2:05F for modulus of elasticity E Fd ¼ 2:15F for modulus of elasticity E > 150 GPa ð24-140bÞ Fd % 2:21F ð24-140cÞ The maximum bending stress at any cross section which makes an angle  measured from the center line of the gap of the ring 150 GPa ð24-140aÞ   12pr2  b ẳ sin2 h 24-141ị 12pr2 h2 max 24-142ị The maximum bending stress which occurs at  ẳ , i.e., at the cross section opposite to the gap of the ring max ¼ The bending stress present in the ring of rectangular cross section in terms of free gap (f ) of the ring, when it is in place in the cylinder b ¼ 0:424f Eh ðd À hÞ2 SI ð24-142aÞ where b and E in N/mm2 h, d, and f in mm The bending stress present in the ring of rectangular cross section in terms of tangential force, F (Fig 2429) b ¼ The bending stress present in the case of slotted oil control ring of rectangular cross section in terms of free ring gap, f bso ¼ 0:424 6ðd À hÞ F ð24-142bÞ wh2 where b in N/mm ; F in N; d, h, and w in mm f Elco Im ðd À hÞ2 Ius 24-142cị where bso in N/mm2 and Ius ẳ moment of inertia of the unslotted cross-section ring, mm4 Im ¼ Ius þ Is Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website ... 25 21 27 23 30 26 15 12 20 20 30 25 35 35 45 40 55 50 60 56 70 60 80 70 85 80 100 85 110 100 120 110 130 120 11 11 11 11 12 12 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13. .. 31 9 4 15 36 0 4 65 404 52 0 451 56 9 481 616 53 3 680 58 8 35 38 50 43 69 60 89 71 111 96 140 122 156 136 176 152 198 171 221 1 93 248 2 15 271 229 2 93 254 32 4 280 27 23 38 33 53 46 68 59 85 73 107 93. .. 13 13 28 26 35 35 52 47 68 60 85 78 1 05 95 110 1 05 1 25 110 140 1 25 150 140 190 150 210 190 210 190 2 25 210 9 10 10 16 15 24 21 28 26 35 31 35 35 40 35 44 40 49 44 63 49 70 63 70 63 75 70 H J K